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Abstract
If A is a subset of the normed linear space X, then A is said to be proximinal in X if for each xÎX there is a point y0ÎA such that the distance between x and A; d(x, A) = inf{||x-y||: yÎA}= ||x-y0||. The element y0 is called a best approximation for x from A. If for each xÎX, the best approximation for x from A is unique then the subset A is called a Chebyshev subset of X. In this paper the author studies the existence of finite dimensional Chebyshev subspaces of Lo.
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References
- Borodin, P.A. Chebyshev subspaces in the spaces L1 and C, Mathematical Notes, 91, Issue 5(2012), 770-781.
- Deutsch, F., Nürnberger, G., Singer, I. Weak Chebyshev subspaces and alternation, Pacific Journal of Mathematics, 1980, 89, 9-31.
- Garkavi, A.L. “On Chebyshev and almost-Chebyshev subspaces”, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 1964, 28(4), 799-818.
- Rakhmetov, N.K. On finite-dimensional Chebyshev subspaces of spaces with an integral matric, Mathematics of the USSR-Sbornik, 1993, 74(2), 361-382.
- Kamal, A. On Copositive Approximation over Spaces of Continuous Functions II. The Uniqueness Of Best Copositive Approximation, Analysis in Theory and Applications, 2016, 32(1), 20-26.
- Deutsch, F. and Lambert, J. On continuity of metric projections, Journal of Approximation Theory, 1980, 29, 116-131.
- Mairhuber, J.C. On Haar's theorem concerning Chebyshev approximation problems having unique solution, Proceeding of American Mathematical Society, 1956, 7, 609-615.
- Singer, I. “Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces” Springer-Verlag, Berlin, 1970.
- Ahiezer, N.I., Krein, M.G. “Some Questions in the Theory of Moments”, Translations of Mathematical Monographs, 2, American Mathematical. Society, Providence, R.I ,1962.
- Cheney, E.W. “Introduction to Approximation Theory”, McGraw Hill Book Company, New York, 1966.
- Kothe, G. “Topological Vector spaces I”, Springer Verlag 1969.
- Semadeni, Z. “Banach spaces of continuous functions I”, PWN, Warsaw 1971.
- Schaefer, H. “Banach Lattices and Positive operators”, Springer-Verlage, Berlin Heidelberg New York, 1974.
- Lacey, H.E. “The Isometric Theory of Classical Banach Spaces”, Springer Verlag, 1970.
- Schoenberg I. and Yang, C. On the unicity of solutions of problems of best approximation, Annali di Matematica Pura ed Applicata. 1961, 54, 1-12.
References
Borodin, P.A. Chebyshev subspaces in the spaces L1 and C, Mathematical Notes, 91, Issue 5(2012), 770-781.
Deutsch, F., Nürnberger, G., Singer, I. Weak Chebyshev subspaces and alternation, Pacific Journal of Mathematics, 1980, 89, 9-31.
Garkavi, A.L. “On Chebyshev and almost-Chebyshev subspaces”, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 1964, 28(4), 799-818.
Rakhmetov, N.K. On finite-dimensional Chebyshev subspaces of spaces with an integral matric, Mathematics of the USSR-Sbornik, 1993, 74(2), 361-382.
Kamal, A. On Copositive Approximation over Spaces of Continuous Functions II. The Uniqueness Of Best Copositive Approximation, Analysis in Theory and Applications, 2016, 32(1), 20-26.
Deutsch, F. and Lambert, J. On continuity of metric projections, Journal of Approximation Theory, 1980, 29, 116-131.
Mairhuber, J.C. On Haar's theorem concerning Chebyshev approximation problems having unique solution, Proceeding of American Mathematical Society, 1956, 7, 609-615.
Singer, I. “Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces” Springer-Verlag, Berlin, 1970.
Ahiezer, N.I., Krein, M.G. “Some Questions in the Theory of Moments”, Translations of Mathematical Monographs, 2, American Mathematical. Society, Providence, R.I ,1962.
Cheney, E.W. “Introduction to Approximation Theory”, McGraw Hill Book Company, New York, 1966.
Kothe, G. “Topological Vector spaces I”, Springer Verlag 1969.
Semadeni, Z. “Banach spaces of continuous functions I”, PWN, Warsaw 1971.
Schaefer, H. “Banach Lattices and Positive operators”, Springer-Verlage, Berlin Heidelberg New York, 1974.
Lacey, H.E. “The Isometric Theory of Classical Banach Spaces”, Springer Verlag, 1970.
Schoenberg I. and Yang, C. On the unicity of solutions of problems of best approximation, Annali di Matematica Pura ed Applicata. 1961, 54, 1-12.